Problem 2. The dancing competition. (15 points)
You and Barbara are the jury members of a dancing competition. After every performance, you and Barbara report an integer grade between 1 and 10, and the final grade for the candidate will just be the average of the two reported grades.
You have just seen the first dancer, and you value her performance at 8, whereas Barbara values the performance at 7. That is, your desired grade is an 8 and Barbara’s desired grade is a 7. Since you are both loud and clear about your opinion, you both know the desired grade of the other person. Now it is time for both of you to report a grade, which may well be different from your desired grade, because the final grade also depends on the other person’s reported grade.
Suppose that the preferences for you and Barbara are as follows: The closer the final grade is to your desired grade, the higher your utility. But remember that Barbara and you have different desired grades.
(a) (5 points) Which choices (reported grades) are rational for you? For ev- ery rational choice, find a belief about Barbara’s choice for which that choice is optimal. For every irrational choice, find another choice, or randomized choice, that strictly dominates it.
(b) (5 points) Which grades can you and Barbara rationally report under common belief in rationality? Which final grades are possible under common belief in rationality?
Barbara was shocked by the large difference in grades you both reported for the first dancer, and it has made her rather insecure. Her preferences have changed because of this unpleasant experience. From now on, her objective is to report the same grade as you. That is, if she reports the same grade as you, her utility will be 1, whereas her utility will be 0 if she reports a different grade than you. Your preferences are similar as before: The closer the final grade is to your desired grade, the better.
The second dancer in the competition is Chris, who danced the jive just like John Travolta in Saturday Night Fever. Your desired grade for Chris’ marvellous performance is therefore a 9.
(c) (5 points) Which grades can you and Barbara rationally report under common belief in rationality? Which final grades are possible under common belief in rationality?